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Full text of "Treatise On Analysis Vol-Ii"

428      XV    NORMED ALGEBRAS AND SPECTRAL THEORY

B = (7+ 7T*)-1 and if for each y e E we put x = B  y, show that the hypotheses
imply that y e dom(T*), by using the fact that x e dom(r*).)

12. Let TV be a normal operator on E. If dom(Nz)  dom(N), show that N is continuous.
(If not, there would exist a closed subspace F of E, such that F is the Hilbert sum of
an infinite sequence (Fn) of nonzero subspaces stable under N and N*, such that the
restriction Nn of N to Fn is a continuous operator and such that

for xn e Fn , where the sequence (an) is increasing and tends to +00.)

13. EXTENSIONS OF HERMITIAN  OPERATORS

(15.13,1) Let E be a separable Hilbert space. A not necessarily bounded
operator H on E is said to be hermitian if: (1) dom(H) is dense in E; (2)
dom(H) c domCff*), and the restriction of H* to dom(H) is equal to H; in
other words, for all x, y in dom(H)9

(15.13.1.1)                     (H-x\y) = (x\H-y).

In particular, (H  x \ x) is real for all x e dom(H).

It should be noted that in general H* is not hermitian. When H is con-
tinuous, this definition agrees with that of (11.5), and a continuous hermitian
operator is the same thing as a continuous self-adjoint operator. An unbounded
self-adjoint operator is always hermitian, but as we shall see below there exist
closed hermitian operators which are not self-adjoint.

Example

(15.13.2) Let Q) be the vector space of indefinitely differentiable complex-
valued functions on R with compact support. If /? denotes Lebesgue measure
on R, then @ may be identified with a subspace of L(R, ft) (13.19), which is
dense in L(R, /?) (17.1.2). It follows immediately from the formula for
integration by parts (8.7.5) and the fact that the functions belonging to ^
vanish outside a compact set, that for all x9 y in ^

= f Dx(t)XT) dt=- f x(ODXF) dt = -(x \ Dy).

(D* | y)

This may be expressed by saying that iD = H is a hermitian operator. The
conclusion is the same if we replace L(R, ft) by Lc(I, /?), where I is a (boundedrem and the